SE Maths - Teaching with Proofs: Expert Tips for Secondary Education

[CaptainBlack]

In online maths fora we often see posts like the first post in thttp://www.mathhelpboards.com/f15/book-recommendations-proofs-1649/#post7709 from kanderson.

When I was in secondary education and then an undergraduate we were never taught about poofs, rather we saw them and produced our own. When I was first at university we were recomended Polya, but it was already too late there was lttle in "How to Solve It" that was not familiar. The only thing I learned about proof after leaving secondary education was a research technique: "If you can't prove what you want, prove what you can" and that I had to manufacture myself to resolve a crisis while writing up my thesis.

So my question is, when did picking up methods of proof become a problem (assuming it is)

CB

 

[caffeinemachine]

In online maths fora we often see posts like the first post in thttp://www.mathhelpboards.com/f15/book-recommendations-proofs-1649/#post7709 from kanderson.

When I was in secondary education and then an undergraduate we were never taught about poofs, rather we saw them and produced our own. When I was first at university we were recomended Polya, but it was already too late there was lttle in "How to Solve It" that was not familiar. The only thing I learned about proof after leaving secondary education was a research technique: "If you can't prove what you want, prove what you can" and that I had to manufacture myself to resolve a crisis while writing up my thesis.

So my question is, when did picking up methods of proof become a problem (assuming it is)

CB

Few years back when I first encountered the problem that "In a party 6 friends meet, prove that one can find 3 mutual friends or 3 mutual strangers among them" I found out that Pigeon-Hole principle lies at the heart of the solution. Although pigeon-hole principle is a pretty natural thing in itself, it was not obvious to me that the problem could be solved using PHP. Then many Olympiad problems had their solutions starting with "Consider the smallest...", and suddenly the problem became trivial. So this was another proof technique I had to assimilate-The Extremal Principle. Induction seemed to be natural in many cases but in the beginning I used to find it tricky 'on what I should apply induction'.
But now, after so many years, I am comfortable with method of proof I see.

 

[kanderson]

Lol sorry, its not that its hard, I just think the books I get are pretty bad at explaining or don't have as much excercises.

 

[Jameson]

Lol sorry, its not that its hard, I just think the books I get are pretty bad at explaining or don't have as much excercises.

I think CB has a legitimate point and wasn't picking on you, just your thread prompted him to make his post. The first time proofs were introduced to me was in 8th grade geometry and they weren't brought up again until Linear Algebra in college. When brought they were more like "watch and repeat" forms, which doesn't lead to a deep understanding. I wish proofs were introduced in high school curriculum to some degree everywhere.

 

[Matrix0]

I understand the importance of proofs in mathematics and their role in developing critical thinking skills. It is concerning to hear that students are not being taught about proofs in secondary education and are instead expected to pick them up on their own. Proofs are a fundamental aspect of mathematics and should be taught at an appropriate level in secondary education. This not only prepares students for higher level math courses, but also helps them to develop logical reasoning and problem-solving skills.

I believe that the lack of emphasis on proofs in secondary education may be due to time constraints and the pressure to cover a wide range of topics in a limited amount of time. However, it is important for educators to prioritize teaching proofs and provide students with the necessary tools and techniques to understand and construct proofs.

Furthermore, as mentioned in the post, it is important for students to be exposed to a variety of proof methods and to be able to adapt them to different situations. This not only helps in understanding the concept of proof, but also prepares students for future challenges in mathematical research.

In conclusion, I believe that teaching proofs in secondary education is crucial for the development of critical thinking skills and a deeper understanding of mathematics. It is never too late to learn about proofs, but it is important for educators to provide proper guidance and resources for students to develop these skills at an earlier stage.